local stability conditions for the babuska method of ......spheres, cv c r", such that for each...

MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 152

OCTOBER 1980, PAGES 1113-1129

Local Stability Conditions for the Babuska Method

of Lagrange Multipliers*

By Juhani Pitkäranta

Abstract. We consider the so-called Babuska method of finite elements with Lagrange

multipliers for numerically solving the problem Au = f in il, u = g on 3Í2, iî C Rn,

7i > 2. We state a number of local conditions from which we prove the uniform stabili-

ty of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm.

The stability conditions given weaken the conditions known so far and allow mesh re-

finements on the boundary. As an application, we introduce a class of finite element

schemes, for which the stability conditions are satisfied, and we show that the conver-

gence rate of these schemes is of optimal order.

1. Introduction. Let Í2 be a bounded simply connected domain in R", n>2,

with smooth boundary 3Í2. We consider the second-order Dirichlet problem

(1.1) Au = f in Í2, u=g on 9Í2.

If /GZ,2(£2) and g G /T(9Í2), r > 1, the problem (1.1) admits the following variational

formulation: find a pair {u, \p), u G //'(fi), i// G ¿2(9Í2), such that

,, „, I Vw • Vuc/x - I v\j)ds + j utpds = - \ fvdx + gyds,(1.2) Jn Jsa Jm Jsi Jdn

for all (v, tp) G //'(ft) x ¿2(3i2).

If (u, i//) is the solution of problem (1.2), then u is the weak solution of problem (1.1)

and \¡j equals du/dn, the normal derivative of u on 8Í2 (cf. [1]).

If Af¡ C //'(Í2) and Aîtj C Z,2(9£2) are finite-dimensional, the Lagrange multiplier

method [1], [2] for the approximate solution of problem (1.1) consists of seeking for

a pair (un, \pn) G Mh x M2 such that

(1.3) SnVu» " Vvdx " k^"ds + knu^ds = ~í¿vdx + kng{pds'

for all (v, «p) G M* x JM*.

It is well known, cf. [1], [2], [4], that the uniform stability of the Lagrange multiplier

method can be achieved only by rather careful choices of the subspaces Mh and M2 .

In particular, if Mh is fixed, then the choice of M2 is critical [2]. In [1] it is shown

that for finite element subspaces the stability can be achieved provided, essentially,

that the diameter of the smallest element on 8Í2 and that of the largest element in the

Received January 31, 1979; revised March 7, 1980.

1980 Mathematics Subject Classification. Primary 65N30.

* This work was supported by the Finnish National Research Council for Technical Sciences.

© 1980 American Mathematical Society

0025-5718/80/0000-01 54/$05.2 5

1113

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

1114 JUHANI P1TKARANTA

interior of Í2 are related as

(1.4) dmin(9í2)>Cdmax(í2),

where C depends on £2.

The problem of weakening the condition (1.4) was studied in [8], where it was

shown that for quasiuniform finite element meshes in R2 the constant C in (1.4) can

be allowed a value close to unity, with certain natural choices of the boundary sub-

spaces. In the present paper we consider the more general situation where mesh re-

finements are allowed. Neither (1.4) nor the conditions given in [8] allow mesh re-

finements on 9Í2.

The technique we use is somewhat different from that of [1 ] or [8]. Instead of

working with the Sobolev space //~'/2(9i2), we carry out the analysis using Z,2(9£2),

supplied with certain weighted, mesh-dependent norms.

In Section 2 a number of technical assumptions are made, which are sufficient

for the uniform stability of the Lagrange multiplier method. The assumptions are given

in a form allowing them to be checked locally for subspaces of finite element type.

In Section 3 the uniform stability of the Lagrange multiplier method is proved

from the assumptions of Section 2. Here the key result, which allows local conditions

in the case of mesh refinements, is Lemma 3.2. The uniform stability yields immediate-

ly an abstract quasioptimal error bound for the approximate solution. As a by-product

we obtain here a weighted convergence result for u - un on the boundary.

The verification of the stability conditions in practice is finally considered in

Section 4, where compatible finite element-type subspaces are constructed assuming a

simplicial mesh in the interior of Í2. Quasioptimal convergence rates are proved for

the resulting finite element schemes.

Among other cases, where the conditions of Section 2 can be verified, we men-

tion the boundary subspaces constructed in [8] and [9] for plane domains.

2. The Basic Assumptions. For a domain S2 C R" we let //fc(i2), k > 0, k inte-

ger, denote the usual Sobolev space with the norm

lK*<n) = E l»l/,n>/=o

where

Mj.ii = X /o^")2 dx' « = (<*i> • • • > O, lal = a, + • • • + a„.|a|=/

For nonintegral k, the norm || • \\Hkm\ is defined by interpolation [7].

For a definition of the spaces //fc(9i2), cf. [1]. To define a convenient norm

for Hk{dü,), when k is an integer, let {Sv} be a finite family of open, disjoint subsets oi

of 9Í2 such that \JV Sv = 9Í2. For each Sv, let {x[v^,. . . , x^ } be a system of local

coordinates, chosen so that Sv has the representation

Sv= {(x<">, ..., X<">) G R» ; x« = Fv{x\v\ ..., xn% ),

{x\v\...,xiv2x)GUv}.


THE BABUSKA METHOD OF LAGRANGE MULTIPLIERS 1115

If the diameters of the subsets are sufficiently small, the local coordinates can be

chosen so that each Fv satisfies

(2.2) \DaFv{x)\ <C, xG Uv,

where C depends only on |a|, not on v. Assuming this is the case, we can define

(2-3) Wilson) - Z WWbHvj.V

where

(2.4) *„<#>, .... xn%) = *(*<">, . . . , x«, Fvix[v\ ..., xW )).

Henceforth we consider a family {r'1(9i2)}0<ft <-, of partitionings of 9Í2 satisfying

some further conditions as given below. We let K and À be real and L an integral pa-

rameter, K > 1, L > 1, X > 0. The subsequent conditions are assumed to hold for

any given h, 0 < h < 1, and for some fixed, finite values of K, X, and L.

Consider a partitioning 7^(9^2), h fixed. For a set S C R", we let d{S) denote

the diameter of S. Our first assumptions are the following:

Al. (i) For all S G ̂ {dSl), d{S) < A.

(ii) To the partition 7^(90) = {Sv} there corresponds a collection {Cv} of

spheres, Cv C R", such that for each v, the center of Cv is in Sv, d{Cv) = SKd{Sv),

and for each C G {Cv}, C n Cv # 0 for at most Z, spheres Cv G {Cv}.

Here Al(ii) may be regarded as a local regularity assumption. We will need an

additional global assumption which restricts the possible global mesh refinements. To

this end, let us define, for a given S G r'I(9í2), the sets 2fc(S) and Rk{S) as follows:

zZ0{S) = S,

(2'5) Zfc(S) = 2Ä_, {S) U {5V G r"(dí2); Sv n 2^(S) * 0}, fc = 1, 2.

and

(2.6) *°(5) = *Äk(5) = Sfc(5)-2^,(5), A: =1,2, ....

Each Sß G Th{d£l) is obviously a subset of/?fc(Sv), 5V G r'"(9í2), for exactly one value

of k. We assume :

A2. If Sv, 5M G t"(9í2) and SM C Rk{Sv), then d{Sß)/d{Sv) <K ■ kx.

In practice, A2 means that the refinements should be globally subexponential.

For situations where refinements of this type are used, cf. [3].

We associate next a family of finite-dimensional subspaces M2 to the partitionings

7^(9^2). For each h, M2 is assumed to have a basis {\px, . . . , \¡/m } which satisfies:

A3, (i) For all /, 1</ < mh, suppty-} C Zk{Sv) for some Sv G /(9Í2) and

k<L.

(ii) If * = 2Zmh ß.^., ß. G /?', and \ = {/; ^ ? 0 on 5v}, 5V G r»(9fi), then

* Z tfiiiMi2W > ii^ii£2(5v) >^_1 Z ^^/iii2(5v)-/6AV /GAV

In addition, we assume the following:

A4. For each Sv G rh(ba), there exists 0V eij such that 5V n supp{0v} # 0,


1116 JUHANI PITKARANTA

d{suM6v})<Kd{Sv),<md

f 9vds2 >K-1d{Sv)"-1 f 62vds.Jdii v v v' Jdii v

Finally, let {Mx}0<h<x be a family of subspaces of//'(Í2). We assume that for each

h the following compatibility conditions hold between Mh and Mh.

A5. To the basis {\px, . . . , \¡/m }, of Mh, there corresponds the set

iui> • • • >%,} cMt suchthat

(i) For any v¡, 1 <i <mn, v¡v- ̂ 0 for at most L functions i>-, 1 <j<mn.

(ii) If5vGr"(9i2)then

{/; v¡ ÍOon 5V} = {/; fy ÍOon Sv} = Av.

(iii) If/ G A-, then

K^Wi.« < llalli2CSv) <^ll^lli2(Sv).

(iv) If i// = H^ß^j, v = l^"&jVj, and Sv G r/l(9í2), then

K Z 732IIÎII2(5V) > IMll2(sv) > X'1 Z ^^ll!2(5v).

and

/ ^*>^-1iiîii2(Sv).

Qualitatively, A5 imposes an upper bound for the dimension of Mh, as compared with

that of the trace space of Mh on 9Í2. In this sense A5 is a condition of the same

spirit as (1.4). The difference is that A5 is only a local constraint.

3. A Stability Result. In this section, Cor C denotes a positive constant which

may take different values on different usages and may depend on n and on the param-

eters K, X, and L of the preceding section without explicit indication. We write C(f2)

if the constant depends also on £2.

If q G /?', denote by ||-\\n dn the weighted L2-norm

V

where the sum is over all Sv G 7,I(9Í2). We will denote by XT„ the space //'(Í2) x

L2(9£2) supplied with the norm

||(«, „)|iiTÄ = l<n + H<-K,9n + Mli._H.an-

In this notation, our basic stability and convergence result is as follows:

Theorem 3.1. Let the partitionings iîSl) and Th{d£l) and the spaces Mh C

//'(Í2) and M2 C £2(9£2) be defined so that the conditions Al through A5 are satis-

fied for some K, X, and L. Further, let fGL2(£2), g G //r(9í2), r > 1, and let {u, \¡j)

be the solution of problem (1.2). Then if h < hn, h0 = /»0(i2) G (0, 1), problem (1.3)

has a unique solution (uh, ¡¡)n)G Mh x Mh which satisfies

IK«*. K)WxTh <C(í2)(||/||0jÍ2 + ltoV_ai8n),



and

\\(u, i>) - (uh, t¡,h)\\x < C min ||(M, M-(v, *)\\x

The proof of Theorem 3.1 will be based on the following two results.

Proposition 3.1. IfuGH1 (£2) is such that

(3.1) j uyds = 0 for alltpGM*,

then if h is small enough, u satisfies

IMI*,_Ä,ao<Cl«llf-.

Proposition 3.2. For any ipGM2 there exists v G Mh such that

Mi,n + IMI*,-%,an < l and í^ds > C|MI*,a,a„-

Before proving the propositions, let us show that Theorem 3.1 follows from them.

For this, note first that we have

(3.2) l!«||L (n) < an) (l<n + H"ll*,-v4,an) ' . « e//'(fi).

This follows, since ||u||A _% 3n > d{ü) /2ll"ll¿2(an) and since

MHim < c(") 0<n + H"lli2(an))1/2. " e//'(i2).

Now define on XTh x X n the bilinear form

B{u, \¡/; v, iß) = I Vw-Vudx- I v\¡)ds+ I uwds,Jii Jdii Jäii r

so that Eq. (1.3) can be rewritten as

(3.3) B{uh, \¡/h, v, *) = -fnfvdx + fdng*ds, {v,v)GMh x Af\.

The bilinear form B is continuous on XTh x XTh '■

(3.4) |8(m, V; v, <p)\ < ||(«, n\xTh IK», *)\\X.h-

Also, using (3.2) we have

:r*

(3.5) ~fnfvdx + Janw* < C(í2) (ll/ll¿2(n) + ^h.-v„m) IK», *>)HxT„-

In view of (3.3) through (3.5), the proof of Theorem 3.1 is complete if we can show

that [2, pp. 186-188]

B{u, \p;v,y)>C>0.

M)Ímhxmh2 (»,^xm* IK«, *)lbrrAIK», *)\\xrh

For (3.6) to be valid in the present situation, the following two conditions are suffi-

cient (see [4]):

uGNfi & f uipds = 0 for all <¿> G AU

(3.7)'dil

l«l?,n > C(l"li,n + H«H*,-vMn)>



and

(3.8) inf sup

I évdsJdii

GM* ^Mh ll^ll*ll4,an(l»lí,„ + ll»llí,-vi,3n)1/2>C>0.

But (3.7) follows from Proposition 3.1, and (3.8) is equivalent to Proposition 3.2.

Thus, if the propositions are true, the proof of Theorem 3.1 is complete. D

Proof of Proposition 3.1. Let V0 be the unit sphere of R"~l, V0 = {x G R""1

|x| < 1}, and let E0 be the cylinder

E0 = {x = (z, x„) G R» ; z G V0, 0 < xn < 1}.

We prove first a preliminary result.

Lemma 3.1. Let S > 0 and let 0 G L2(V0) be such that

\Lo(z)dz1 = nL2(V0) <8-

Then if u G //' (E0), there is a constant C depending only on 8 such that

L wz->Vo

0)|2 dz < c\k Vu\2dx + j;Po 6(z)u(z, 0)dz

Proof. If the assertion is not true, then there exist the sequences {uv} and {6V},

uv G //»(E0), 6V G L2(V0), v = 1, 2, ... , such that

(3.9)

(3.10)

(3.11) j;Po

1 =

Wufdx +

vllZ.2(p0)

II«

<5"'

vllf/l(E0)= 1,

LAW2- o)& 2 <i/pnK(z, o)i2dz.

By (3.10), there exists a subsequence of {"v}, still called {uv}, which converges in

L2(E0). Since (3.10) also implies that /p \uv(z, 0)|2 dz < C, we have, by (3.11), that

¡EQ\Vuv\2dx-*0, v->°°.

Hence, {uv} converges in //'(E0) to a function w0(x) = CQ = const. By (3.10), C0 =^0.

However, taking the limit in (3.11) one has

0 = lim ÍVoev(z)uv(z,0)dz

> Hm \\C0\ L(^dz cwA^po) l|uv Y0ll/Vl(£0)

so that C0 = 0, by (3.9), a contradiction. D

Now let Th(dSl) and Ai2 be given, h sufficiently small so that the subsequent

assumptions are valid. For each Sv G r;,(9í2), let 0V G Mh be a function referred to in

A4, and let the sphere Cv C R" be as in Al(ii). By A4, if h is small enough, there

exist the systems of local coordinates {x^, . . . , *^}, the spheres Vv G R"~l with

center at the origin and diameter d(Vv) - 2Kd(Sv), and the functions F defined on



Vv such that Sv C Tv and supp{0v} C Tv, where

Tv = {(x<v>, . . . , x«) G Ä"; »W = Fv(x("), . . . , xM), (xf>, . . . , xW)G Pv}.

We may further assume the coordinate systems to be chosen so that each F satisfies

\fTFv(z)\<C, |a|=l,zGpv,

and

(3.12) |Fv(z,) - Fv(z2)| < Kd(Sv), z,, z2 G Vv.

For each v, define the set Ev as

Ev = {(x<v>, . . .,x^)GR";{x[-\ . . .,^,)ePv,

Fv(x<"), . . . , jrW) < *W < Fv(x<"\ . . . , xM) + Kd{Sv)}.

Then if /z < h0, h0 = h0{il) sufficiently small, we have Ev C £2 for each v. From

(3.12) and from Al(ii) it is also easily verified that

(3.13) EvCC„n£2, y =1,2, ....

The set Ev may be mapped onto the cylinder E0 of Lemma 3.1 by the mapping

x —► x = Jv{x) defined as

il = jr1d(Sw)-,xp>, 1 = 1.«-1,

x„ = K-'diSX1 [4V) - ^(^iv)- • • • » *&)! •

By the properties of Fv assumed above, Jv: E„ —► E0 is an invertible smooth trans-

formation. Moreover, we obtain from the definition of Jv the formula

(3.14) f rfs)& = f ev(z)^v(z)dz, * G L2(rv),'»O

where $v(z) = ${JV ' (z, 0)) and ev satisfies

(3.15) C,Â-"-'ci(5v)"-' < ejiz) < Ç/"-1*/-', z G V0.

Now let u G H1 (Í2) be such that (3.1) holds, let uv be the function u in the coordi-

nate system {x(,v), . . . , x<v)}, and let ûv(x) = uv(J~1(x)), x G E0. Then, since Tv D

supp{0 }, we have, using (3.1) and (3.14),

(3.16) 0 = L uvevds =L ev(z^v(z)ùv(z, 0)dz,1 v v0

where 0v(z) = 0v(/-'(z, 0)). By A4 we have

Jr.8"*or

Le»6„dz

>K-ld{Sv)"-1f \d/ds,1 V

>K-1diSvy~1 f ev\dv\2dz,•* Tin

or, using (3.15),

(3.17) U:^dz

'Po

>cfVo\ev8vl2dz.



Let 0vbe normalized so that lkv0v||L ,p ^ = 1. Then, choosing 0 = ej)v in Lemma

3.1, we conclude from (3.16) and (3.17) and from the lemma that

f \Ûv{z, 0)|2„z<cf \DlÛv\2dx,JVn J to

or, in the original coordinates,

d(Sv)~l(r \»v\2ds<cf \Dlufdx.

We sum the last inequality over v and use (3.13), together with Al(ii), to finally obtain

ll<-VMn<Z<W-1l Kl2*v lv

C^f \Diuv\2dx<CLJ \Dlu\2dx.

V

V

This completes the proof. D

Proof of Proposition 3.2. Let Vh denote the subspace of ¿2(9£2) spanned by

the functions uf-|an, i = 1, . . • , m„, vt as in A5. It is easily verified from A5 that

dim(^7',) = mft, i.e., V* and M2 are of equal dimension. We also recall from A5(i),

(iii), and (iv) that if v = 2™*/^-, then

mn

l»li,n<£Z ßfWliii^LKZd{Sv)-] Z f3/ll»/llí2(5v) <LK2 lK._tt.an-

v /_AV

In view of this, the assertion of Proposition 3.2 follows if we can show that

(3.18) inf sup _/»«***_ >C>0.*&Mh2 v^v" IMI*,v4>anll»H*,-y4,an

The remaining problem of proving (3.18) can be further modified as follows. First, de-

fine the linear mapping Ph : ¿2(9i2) —► Af2 as

(3-19) PjeAl*,: b(Ph^, v) = b(<//, v) VvGV",

where

(3.20) b(4>,*)= fm^ds, V,<peL2(9i2).

Then we have

Lemma 3.2. (3.18) holds, provided that Pn satisfies

(3.21) lK?*vll*,tt.an<c1Mk%,an. *e_2(dfi).

/Voo/ Denote by //, and //2 the Hubert spaces obtained by supplying L2(9£2)

with the inner products

».irl^LW, and i4>,<p)H=Zd(Sv)~lfv^ds,1 v ■''v l v Ji>v

respectively, where the sums are over Sv G r/l(9í2). Then \\-\\H = ll"llftji/2>3i2, and



INI// - II'llft _i/2 3ii, and b is a bounded bilinear form on //, x H2:

(3.22) Ib^^KIWI//, IMI//2> ^GHx,yGH2.

We also easily contend that

(3.23) inf sup W'*> = inf sup *»»_} = 1.,/,£//, *_//2 II^||//iIMIh2 *eH24,eHl \W\HlW\\H2

Assume that (3.21) holds. Then, using (3.23), (3.19), and (3.21) we find that if

»G r*,then

bQM) Slin b(F^,»)

IMI"2 = Ä ^7= A nwi^r

(3-24) <Csup ^M<Cs«p «£*, .EF».

Proceeding as in [2, p. 113], we now define the linear mapping En : Vh —► Ai2 through

the formula

bU),v) = M,Ehv)Hi, ^GM^vGV*.

Then, by (3.22) and (3.24), F,, is bounded and 1:1 [2, pp. 113-115]. Moreover, as

Ai\ and V" are of equal dimension, Eh is onto Ai\. Then if E*\ Af!¡ —> Vh is the

adjoint mapping defined as

bOM) = (F*iMW ipcMÍ^ev*1,

we have, using (3.24) and a well-known argument (cf. [2, pp. 112-115] or [4]), that

» „_r> HvVIL.il»!!-- ^ *inf „ sup„ h mi i. i = iK^TMr1 = w.

*_Af* uer" ll*H//, H»l„2

= inf sup _________ >C>0.uer" *_*/* ll^llz/jllyll^

This proves the lemma. D

To prove (3.21), let ¡p G _2(9Í2) be given and define, for Sv G r'I(9í2), the func-

tion *pv as

<Pvix) = vix), xGSv,

= 0, x G 9Í2\S„.

With the sets Rk{Sv), k = 0,1,2, ..., defined as in (2.6), let us first prove that

Pnyv{x) decays exponentially in the following sense:

(3.25) IIVvl_2(**(_v)) < Ce~ak^vh2(Svy * > 0,

where a does not depend on h.

Note that, in the case of a quasiuniform partitioning, a result analogous to (3.25)

was proved in [6] in a different context.


1122 JUHANI PITKÄRANTA

We start the proof of (3.25) by first introducing a real parameter q, q > 1, and

defining

(3.26) Vj{x) = qmivj{x), j = 1, . . . , mh,

where m- = 0 if supp{u} n Sv i= 0, and otherwise m- is chosen so that supp{u} n

Sm_j(5v) = 0 and suppfy} n _■„.(£„) ¥= 0, where the sets -fc(5„) are as in (2.5).

Consider a given subset Sß G r/l(9í2) such that Sß C /?,.(_„), fr > 1. By the as-

sumptions A3(i) and A5(ii), if v¡ ̂ Oon Su, then suppfy} n 8__ C _L(_M).

In view of (2.5) and the above definition of m,, this implies that fr - _ < »i^ < fr,

and therefore, by (3.26),

(3.27) Sy(x) - qkvj{x) = qkAjvjix), |A;| < 1 - <TX.

Now let Pn<pv be expanded as Pnyv = I.™* /3/i///, and let u = Z™'' ß£,. Then if AM is as

in A5(ii), we get, using A3(ii), A5(iii), A5(iv) and (3.27),

q-k$siPh*v)ïds

>[K-1-Wl-q-L)]\\Ph*v\\l2(Sß>

(3.28) -2l^)í4A'Y!

>[K-i-m-Q-L)]\iph^\\i2(Sß)

-M{l-q-L) Z ßfL vfds/-A, JH*

> [K-* - m3 + o (i - q-l)] iu**wii_2(J|l).

Taking q sufficiently close to (but still larger than) unity, we can have

(3.29) A-' - K{K3 + 1) (1 - q~L) > 5__-'.

Upon summing over ju, (3.28) and (3.29) then imply

UPh^ds=ZfRUSAPh,v)vds(3.30)

On the other hand, we find from the definition of v and from A3(ii), A5(iii), and

A5(iv) that

/„/v»*<^3/2|l^ll_2(_v)lli>*^ll_2CSv).



Recalling that v G Vh,we then have, using (3.19) and (3.20),

hii{p^)~vds = hii^ds = 5syds

(3.31) <^3/2IIÎIl2(_v)II^ÎIl2(Sv)

<^3/2iWU2WjZ«*^*îii2(_k(.v))î,/2-(k \

Since q > 1, the estimate (3.25) follows from (3.30) and (3.31).

Combining (3.25) with the assumption A2, we get for all Sv, Sß G r"(9£2) the

estimate

d{Sß)f {Pn,v)2ds<d{Stl)j iPh,v)2ds

(3.32)

< CkKe-akd{Sv) f ^vds < C,e-ai *_(_„) f ^ ds,

where A: is such that SM C /?fc(_v), and 0 < a, < a.

Finally, to estimate Phy, write Pny = _v Ph<pv, and apply the inequality

¿a*V <C¿><*a2, /3>0>C=C(P),akeÄ1>i / i

together with (3.32) to first get

<KSu)i iPh*?ds'sß

d(su)fs \l( Z J>v)12

ds

<c_(sM)E^*/-( Z w"k M\ vGAkfc,M

<C1d(_M)ZJV-*,M^fc Z /_ iP^fdsk "6Afc,M M

<C2Ze^)kNKfl Z <«/ ¿A,

where Afe = {v; 5V C Rk{Sß)}, and A^M denotes the number of sets 5V G r'I(9í2)

which are contained in Rk{S ).

Nk can be estimated as follows. First, if Cv is the sphere associated to Sv as in

Al(ii), then, by Al(ii) and A2, the set UvGAfc Cv is contained in a sphere of radius

(3.34) p < Kd(Sß) £ jx + 4K{Kd{Sß)kk) < Cd{Sß)kx+ '.

/-o

Second, by Al(ii) and A2,

(3.35) d{Cv) > 8/q/r'fc-\f(SM)], v G AfcM.



Therefore, using (3.34), (3.35), and Al(ii), we have the estimate

_ [Cd(Sß)kx+ ' ] " > NKtl [4k~xd(Sß)] ",

so that

(3.36) NKß < Ckn(2X+l).

Upon taking 0 < ß < a, in (3.33) and using (3.36), we get

¿(Vf (/»2ds<C2>-"2* £ d(Sv)f <p2ds,JSß k veAfc>M JSv

with a2 > 0. Summing this over p. and again using (3.36), we finally have

Hfy<„,ao <<?_-__ Z e-«2*d(Sv)f Jdsv k MeAkjV ^

= cz(z^2XvV(5v)/s^2*

<ci__( Ze-<^r<2X+1>)d(sv)f *2dsV \*=1 / "v

<c_IMI*,*&,an-

This proves (3.21), and the proof of Proposition 3.2 is complete. D

We point out that the regularity properties of 9Í2 did not play any explicit role in

the proof of Proposition 3.2. Also, Proposition 3.1 is true under weaker regularity as-

sumptions on £2; for example, it would suffice to assume that £2 is a Lipschitzian do-

main with piecewise smooth boundary. Theorem 3.1 is then true also for such more

general domains.

4. Application. Let {Th(dÇÏ)}0<h<x be a family of partitionings of 9Í2 satisfy-

ing assumptions Al and A2. Let h be fixed sufficiently small and assume that for each

Sv G Th(d£l) the systems of local coordinates {x^v\ . . . , xj^} are defined so that Eqs.

(2.1) through (2.4) are satisfied.

Let k be a fixed integer, k > 1. We define Ai2 as the maximal subspace of

_2(9Í2) such that if \p GAf!¡, Sv G /(9Í2), and i//v is defined as in (2.4), then \pv is

a polynomial of degree k - 1 in the variables xy\ . . . , x„_,.

To define the interior subspaces Mh, consider a family of partitionings it" of R"

into n-simplicial subdomains tv. We assume the partitionings to be arranged in the

usual way so that any face of any t G -nh is also a face of another tv Gn". The fol-

lowing additional assumptions will be made:

Bl. (i) rGTT* =*_(*)<*.

(ii) For each t G it", either rn£2 = 0orinf2 contains a sphere of radius p >

K-'d(t).

(iii) If t G n", S G r"(9í2), and t n S # 0, then K~ld(S) < d(t) < Kd(S).



We note that if t G Í2, then (ii) reduces to the ordinary regularity assumption

[5].We let {7-/l(_2)}0<ft< j be a family of partitionings of Í2, defined as

r-*(í2) = {Tv, tv = rM n n, r„ e »*, f„ n n # 0).

We will further denote by f2h the polyhedron

fvejr«;fvnn#0

For each h, let M^ be defined as the maximal subspace of //'(Í2) such that, for each

T G th(Ú), Aíh I- equals the space of polynomials of order k on T, where k has the

same value as in the definition of M2 .

In order to be able to verify the conditions A3 through A5 for the above sub-

spaces, a further constraint on the partitionings will be required. In particular, A5 is

not generally true unless the partitioning r',(9i2) is locally by a certain factor coarser

than the partitioning induced by Th(ÇÏ) on 9Í2. A sufficient condition can be found

as follows. Let {y,, . . . , y } be the set of vertices of the simplices tv G ir" which

are located on d£lh. For each i, 1 < i < ph, let

<4J> 4 = M, *-'

where A;. = {v; tvGirh,yiGfv}. Then it suffices to assume

B2. Each Sv G t"(9í2) contains at least one of the sets 9Í2 n Fft, 1 < i < ph.

To verify the assumptions of Section 2, we need first a basis for M2 . For Sv G

7J'(9Í2), let Uv be as in (2.1), and let CQ > 0 be a fixed constant. Then, by a scaling

argument, there exists for each v a basis {</^v\ . . . , ¡p*$} of the space of polynomials

of degree < fr - 1 in R"'1 such that if V¡ C F"_1, / = 1, 2, 3, are any spheres satis-

fying V; n Uv # 0 and C0d(Sv) > d(V¡) > C¿ld(Sv), then

c-iz^iiaii2(p1)</p2(£^>)2d5(4.2) i=1 \'=1 /

m

<cE^ii^v)iii2(p3v n,^1.i-i

where C depends only on CQ, k, and n. Assuming such bases to be defined, we take

for the basis of M* the set \JV {^»>, . . . , ^}, where

^v)ix) = 4\x\v\...,xn%), xGSv,(4.3)

= 0, x G 9Í2\5V.

For the above basis, A3(i) is true with L = 0. To verify A3(ii), note that, by B2,

Bl(ii), Blfiii), (2.1), and (2.2), if h is sufficiently small, then each Uv in (2.1) contains

a sphere of radius p > Cd{Sv), C> 0, C independent of v. Denote this sphere by


1126 JUHANI P1TKARANTA

Vv ,, and let Vv2 be a sphere of radius p = d(Sv) such that Vv 2 3 Uv. Then the va-

lidity of A3(ii), for some constant K, is obvious from the following chain of inequali-

ties:m m

i=i ;=1

m

>c2zß2yr%2(vv2)>c3i=i

EMí=i(v)

¿2(Pv,2)

> Cm

EWí=i(v) > c,

¿2(5V) 5 IW(v)

>

¿2(Pv,l)

c6£ft2Mv)ni2(pvl)>c7Z^Mv)iii2(pv2)i=i /= i

77!

> c^ßlufhl^y ßi^R1-í=i

Here the constants C are positive and independent of v\ the second, third, sixth, and

seventh inequalities follow from (4.2) and the remaining ones from (4.3), (2.1), (2.2),

and from the inclusions Vv , C Uv C Vv 2.

Assumption A4 is easily verified. Taking

0v(x)=l, xGSv,

= 0, x G 9Í2\5V,

it suffices to verify that, for all Sv G r*(9í2),

f ds > cd(svr~i,

with C independent of v. This follows from B2, Bl(ii), and Bl(iii).

Finally, A5 should be verified. To this end, let {£,, . . . , % } be the set of con-

tinuous functions in R" such that for each i, 1 < /' <p/J, %. is a polynomial of degree

one in each tv G it", \\%¡\\l ,Rn\ = 1, |,-(x) > 0, and supp{|¿} = Eh, where F?1 is as in

(4.1). As is well known, such functions exist. Then define

(4.4) uf (x) = |iv(x(">, . . . , x„v)Vf(xiv), . . . , xW ), 1 < / < m, x G fl.

where {x^, . . . , xnv^} and {y[v\ . . . , f^} are as above, and iv is chosen so that

(4.5)

Functions vjv' are continuous and piecewise polynomials of degree k, so vjv^ G Afh.

Now let us verify that the set \Jv{v(xv\ ■ ■ ■ , v$} is what we are looking for. First,

A5(i) follows from the assumed regularity property Bl(ii). Second, A5(ii) is true since

supp{u^} n 9Í2 C Sv = supp{i//^}. Third, A5(iii) follows from a scaling argument,

using Bl(ii) and (4.2).

Finally, to verify A5(iv), note that if h is small enough, then (2.1), (2.2), Bl(ii), and

Bl(iii) imply the existence of the spheres Vv CR"~l, with d{Vv) > Cd(Sv), C independent

of v, such that £>„ C t/„ and so that %; satisfies

S„ DE" n 3Í2.



(4.6) |/y(x) > C> 0, x G Av = {(X<v>, . . . , je«) G 9Í2; x<v>, . . . , „£_, G Pv}.

Then, since £(. is everywhere nonnegative, we get, using (2.1), (2.2), (4.2), and (4.4)

through (4.6),

fSvitßj^)(tßj^y> í/iv(p/f)2^

>cfVv(tßi^2dx^...dx^1

>ClSUv(tßi^2dx^...dx^i

>c2ÍSv(Ew¡v)yds> ßieRi-

The remaining inequalities of A5(iv) follow from a similar reasoning.

Having verified that the assumptions of Theorem 3.1 are satisfied, let us evaluate

the error of the approximation in terms of the parameter h of Al(i) and Bl(i). We

need the following approximation results:

Proposition 4.1. Let Aih and M2 be defined as above, with k denoting the de-

gree of polynomials in Mh, and let u G /¿* + 1(í2) and \¡j G //fe + '/i(9í2). Then, if

k> (n - l)/2, the following estimates hold

(4.7) min f|« - u|1>n + \\u - v\\h^m} < Chk\\u\\Hk+i

and

(«)'

(4.8) ^W -^.„.anï < Ch II*IW*-%(b„).

Proof. Denote by Nh the finite element space associated to the partitioning of

ilh such that Aih = Nn \n. Further, let ue denote the extension of u to Q.n such that

(4-9) H"ell/Y*+i(_„) < C\\u\\Hk+Hny

Such an extension exists since 9Í2 is smooth.

Let Ihue G Nh denote the Äth-order Lagrange interpolant of ue on £2ft. If k >

(n - l)/2, Ih is a continuous linear operator on Hk + l(Q,h), and, by the classical results

of approximation theory (cf. [5, p. 121]),

(4.10) Hi/ -Ihu<II//i(lïft)< Chk\ue\k+xMh.

To estimate \\ue - Ihue\\h _,h m, let tv G tt" be given such that fv n 9Í2 =£ 0, and let

/ denote the scaling mapping

Jv-x^x = d{F;y

Further, let Jv(tv) = tv, Jv(tv n 9S2) = fv, ûe(x) = ue(x), and îhûe(x) = ¡hue{x),

xGt .



By the assumption Bl(ii), if h is small enough, the estimate

L(ûe-îhûe)2ds<c\\ûe-îhûe\\2HH%)1 V

holds uniformly for all iv = Jv(tv), tv G -n", tv n 9Í2 =£ 0. By the approximation

properties of Ih we then have [5]

f%iû'-îhû*)2ds<cm2k+h?v.

Upon scaUng this back to the original size we get

(4.11) d{tvrllv^niue-iy)2ds<cd{tv)2k\ue\2k+utv.

Summing over v and using Bl(i) and Bl(iii), (4.11) gives

(4.12) ||>/ - Vell*,-V4,an < ^11"%*+>("„)•

In view of (4.9), (4.10), and (4.12), the estimate (4.7) follows by choosing v = I„ue\n.

To prove (4.8), let \¡/ G //fc(9£2) and consider a given partitioning Th{d£l). For

each Sv G t'"(9í2), let {x^\ . . . , x^} be as in (4.3), and assume that h is small enough

so that (2.1) through (2.4) hold. For each Sv G Th(bü), let Vv C F""1 be a sphere

such that Vv D Uv, where Uv refers to (2.1), and

(4.13) d{Vv) < Cd{Sv),

for some constant C. Further, let

Tv = {(x<">, . . . , *W) G 9Í2; („M, . . . , x<l, ) G flv},

and let i//v be defined on Vv as

*v(x<*\ . . . , *M i ) = ^(„(v), . . . , -0»)f (-(v)) . _ > x(v)} e Fv

In view of (4.13), there exists for each v a polynomial ipv on fv of degree k - 1 such

that

(4.14) Mv-<Pvh2ÍVv)<CdiSv)k\tl,Xvv>

where Cis independent of v (cf. [5, p. 115]). On the other hand, by the assumption

Al(ii), the constant C in (4.13) can be chosen so that, for any given T G {Fv}, F inter-

sects Tv for at most L different Tv G {rv}. Therefore, if h is small enough, (4.14),

Al(i), and (2.3) imply

E<«ii*v - *vii_2(pv) < ^2*+1Ei^iU<ciA2*+1iW'Wi)-V V

In view of this we have

<4-15) ¡"jM*-dU,„,8n <chk+*W\Hk(my



The estimate (4.8) now follows from interpolation between (4.15) and the trivial esti-

mate

min 110 - y \\h y. a <hv*W\\L (an).i£eAf2 z

This completes the proof. D

Using Proposition 4.1 one can now easily estimate the convergence rate of the

Lagrange multiplier method.

Theorem 4.1. Let u be the solution of problem (1.1) and let {uh, \¡jh) be the

solution of (1.3), with Mh and Ai2 defined as above, and so that Al\ contains the

polynomials of degree k. Then iffGHk~i (£2) and gGHk+'Á(d£l) in (1 A), and ifk >

(n - l)/2, the following error bounds hold:

"' Ei) < CÄ*(||/||ff„_i(n) + l_1lff„+a(3n)),XTh

and

u - uhh2(ii) < Chk+i(\\f\\Hk-Hn) + \'ig\\Hk+V2(dn)).

Proof. The first inequality follows from Theorem 3.1, Proposition (4.1), and the

estimate (cf. [7])

IMI//fc+i(n)+Hk~\zil)

< C{\\f\\Hk-Hn) + \\g\\Hk+vK,n)), k>l.

The second estimate follows from the ordinary duality argument, see [1]. □

Institute of Mathematics

Helsinki University of Technology

SF-021S0 Espoo 15, Finland

1. I. BABUSKA, "The finite element method with Lagrange multipliers," Numer. Math., v. 20,

1973, pp. 179-192.

2. I. BABUSKA _ A. K. AZIZ, "Survey lectures on the mathematical foundations of the

finite element method," in The Mathematical Foundations of the Finite Element Method with Ap-

plications to Partial Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1972.

3. I. BABUSKA, R. B. KELLOGG & J. PITKÄRANTA, "Direct and inverse error estimates

for finite elements with mesh refinements," Numer. Math., v. 33, 1979, pp. 447—471.

4. F. BREZZI, "On the existence, uniqueness and approximations of saddle-point problems

arising from Lagrange multipliers," RAIRO, Sér. Anal. Numèr., v. 8, 1974, pp. 129-151.

5. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland,

Amsterdam, 1978.

6. J. DOUGLAS, JR., T. DUPONT & L. WAHLBIN, "The stability in iß of the /^-projec-

tion into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193—197.

7. J. L. LIONS & E. MAGENES, Problèmes aux Limites Non Homogènes et Applications,

Vol. 1, Dunod, Paris, 1968.

8. J. PITKARANTA, "Boundary subspaces for the finite element method with Lagrange

multipliers," Numer. Math., v. 33, 1979, pp. 273-289.

9. J. PITKÄRANTA, 77ie Finite Element Method with Lagrange Multipliers for Domains With

Corners, Report HTKK-MAT-A141, Institute of Mathematics, Helsinki University of Technology,

1979.


local stability conditions for the babuska method of ......spheres, cv c r", such that for each...

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